Johansen and Juselius (1990)

OXFORD BUtXETlN OF ECONOMICS AND STATISTICS. 52,2 (1990) 0305-9049 S3.00

MAXIMUM LIKELIHOOD ESTIMATION AND INFERENCE ON COINTEGRATION - WITH APPUCATIONS TO THE DEMAND FOR MONEY Soren Johamen, Katarina Jtiselius I.

INTRODUCTION

LI. Background Many papers have over the last few years been devoted to the estitnation and testing of long-run relations under the heading of cointegration. Granger (1981), Granger and Weiss (1983), Engle and Granger (1987), Stock (1987), Phillips and Oullaris (1986), (1987), Johansen(1988b), (1989), Johansenand Juselius (1988), canonical analysis. Box and Tiao (1981), Velu, Wichem and Reinsel (1987), Pena and Box (1987). reduced rank regression, Velu, Reinsel and Wichem (1986), and Ahn and Reinsel (1987), common trends. Stock and Watson (1987), regression with integrated regressors, Phillips (1987), Phillips and Park (1986a), (1988b), (1989), as weU as under the heading testing for unit roots, see for instance Sims, Stock, and Watson (1986). There is a special issue of this BULLETIN (1986) dealing mainly with cointegration and a special issue of the Journal of Economic Dynamics and Cotitrol (1988) deeding with the same problems. We start with a vector autoregressive model (cf. (1.1) below) and formulate the hypothesis of cointegration as the hypothesis of reduced rank of the longrun impact matrix II = afi'. The main purpose of this paper is to demonstrate the method of maximum likelihood in connection with two examples. The results concern the calculation of the maximum likelihood estimators and likelihood ratio tests in the model for cointegration under linear restrictions on the cointegration vectors 0 and weights a. These results are modifications of die procedure ^ven in Johansen (1988b) and apply the multivariate technique of partial canonical correlations, see Anderson (1984) or Tso (1981). For iiêrence we apply the results of Johamen (1989) on the asymptotic distribution of thelikelUuKxl ratio test. These disttibutiom are givai in terms of a multivmate Brownian motion process and are tabidated in the Appendix. Inferences on a aiydfiimder linear restrictions can be amducted using the usual x^ distribution as an approximation to the distribution of likelihood ratio test. We also apply the limiting distribution of the tnaximum liketifaood estimator to a Wald test for hypotheses about a and 0. 169

170

BULLETIN

I.I The Statistical Model Consider the model H,:X,=n,X,_,-l-... + ntX,_,-l-^ + 4»D,+e,,(/ = l,...,r),

(1.1)

where £,,...,6^ are IINp(O, A) and X_ji.n,...,Xo we fixed. Here the variables D, are centered seasonal dummies which sum to zero over a full year. We assume that we have quarterly data, such that we include three dummies and a constant term. The unrestricted parameters (^, * , II,,..., n;t, A) are estimated on the basis of T observations from a vector autoregressive process. For a /ndimensional process with quarterly data this gives Tp observations and /> -I- 3p + Arp^ +/'(p +1 )/2 parameters. In general, economic time series are non-stationary processes, and VARsystems like (1.1) have usually been expressed in first differenced form. Unless the difference operator is also applied to the error process and explicitly taken account of, differencing implies loss of information in the data. Using A = 1 - L, where L is the lag operator, it is convenient to rewrite themodd(l.l)as AX, = r,AX,_| + ...-l-rk_,AX,_4 + i + IIX,_t-l-/( + *D,-l-e,,

(1.2)

where r,= - ( i - n , - . . . - n , ) ,

{i=i,...,k-\),

and Notice that model (L2) is expressed as a traditional first difference VARmodel except for the term IIX,_^. It is the main purpose of this paper to investipte whether the coefficient matrix II contains information about long-run relationships between the variables in the data vector. There are t h r ^ possible cases: (i) Rank(Il)=p, i.e. the matrix II has full rank, indicating that the vector process X, is stationary. (ii) Rank(Il)=0, i.e. the matrix 11 is the null matrix Mid (1.2) corresponds to a traditional differenced vector time series modd. (iii) 0 < rank(n) = r < p implying that there are p x r matrices o and fi such The cointegration vectors fi have the property that fi'^, is stationary even though X, itself is non-stationary, in this case (1.2) esR be interpreted as an error correction moctel, see Ettê and Granger (19S7), Davidson (1986) or Jcdiansen {1988a). Thus the main hypothesis we sttl£ consider here is the hjpotlîs of rcoint^-ati(Hi vectors Hj:n=a^', where o and jP are p X r matrices.

:

(14)

INFERENCE ON COINTEGRATION

171

We further invest%ate linear hypotheses expressed in terms of the coefficients fi, a and fi, and in particular the relation between the constant term and the reduced rank matrix II. If D is restricted as in H^, see (1.4) and / ( # 0 the non-stationary process X, has linear trends with coefficients which are functions of/J only through a\fi, where a^ is a/? x(j? - r) matrix of vectors chosen orthogonal to a. Thus the hypothesis /i = aP[f, or alternatively o V /* = 0, is the hypothesis about the absence of a linear trend in the pr(x;ess. Note that when ft = a^(, we can write where /?* = (/»',fi'o)'and Xf_t==(X;_4, 1)'. This is useful for the calculations. Since the asymptotic distributions of the test statistics and estimators depend on which assumption is maintained, it is important to choose the appropriate model formulation. This has been pointed out for instance by West (1989), Dolado and Jenkinson (1988). The mathematical results for the multivariate model (1.2) are given in Johansen (1989). }.3. The Data We have chosen to illustrate the procedures by data from the Danish and Finnish «;onomy on the demand for money.' The relation m=f{y,p,c) expresses money demand m as a function of real income y, price level p and the cost of holding money c. Price homogeneity was first tested and since it was clearly accepted by data the empirical analysis here will be for real money, real income and some proxies measuring the cost of holding money. Money, income and prices were measured in logarithms, since multiplicative effects are assumed. The two data sets differ both as to which variables are included and the length of the sample. More interestingly, however, the institutional relations in the two economies have been quite different in the sample period. In Denmark, financial markets have been much less restricted tfeui in Finland, where both interest rates and prices have been subject to regulation for most of tiie sample period. One would expect this to show up in the empirical results and it does. For the Danish data the sample is 1974.1-1987.3. As a proxy for money demand ml was chosen because the data available on a quarterly basis are based on more homcêneous defimtions for ml than for m l . The cost of holding money w ^ assiimed to be approximately measured by the difference between the bank deposit rate, i'', for interest bearing deposits (whidi are the main part of ml) and the bond rate, i'', which plays an important role in the Danish &xmomy. The two interest rat^ were included unrestrktedly in the aimlysis, but subsequently tested for equal coefficients with o{^K)site sipis. The inflation rate. A/?, was also inclwled as a poîble proxy for tt^ cost of ' For a general review erf theoretical aed emprical results on the demand for money, see for instance LakBCT(1985).

172

BULLETIN

holding money, but since it did not enter siîificantly into the cointegration relation for money demand it was omitted from the present analysis. For the Finnish data the sample is 1958.1-1984.3. In this case ml was chosen since the m 1 cointegratitMi relation was found to enter the demand for money equation more significantly and hence illustrated the methodology better. Since interest rates have been regulated, a good proxy for the actual costs of holding money is difficult to find. The inflatk>n rate, Ap, is a natural candidate and therefore is included in the data set. Moreover, the marginal rate of interest, /'", of the Bank of Finland is included in spite of the fact that the marginal rate measures restrictedness of money rather than the cost of holding money. It has, however, been chosen as a determinant of Finnish money demand in other studies and therefore is also included here. AU series are graphed in Figure 3 and Figure 4 in Section IV. The data are available from the authors on request. The p^jer is structured as follows: Section II discusses the various hypotheses we shall investigate and in Section III the notation is introduced for the maximum likelihood procedure. The next section derives the estimates of a and fi under-the assumption of cointegration and the last two sections investigate estimates and tests for fi and a under linear restrictions. Throuôut, the two examples are used to motivale the statistical analysis and to illustrate the mathematically deriv^l concepts. n.

A CLASSIFICATION OF THE VARIOUS HYPOTHESES

The hypotheses we consider consists of the hypothesis Hj on the existence of cointegrating relations combined with linear restrictions on either the cointegrating relations or their weights: '(or j8 =

and Hf is /fy augmented by/* = oôfor/= 2,...,5. Note that the hypothesis H,, where II is unrestridKl, can be written as Hj with r=p. Hence, in this case the restriction /i - afi'g is the same as having /t unrestricted. When we estimate model (1.2) wider the hypothesis n = a^' the choice of hypothesis about fi becoRKs important. Ffew the Danish data tfiere does not seem to be any linear trend in the non-^tiooary processes (cf. Figure 3) and we will estimate modek of the form Hf. For the Finnish data, however, there seems to be a linear trend in the non-isationary processes (cf. Figure 4) and models of ttie form/f, will be ^timated. The matrices A(p x m) and H{p x .y) are known emd define linear restrictions it is significant, and the hypothesis i/ji^) is maintained. We find ^ as the first three columns of ^ from liable 2 and d as the corresponding columns of the weights W. Note that given the full matrices V and VV one can estimate a and P for any value ofr. For the case r>\, the interpretation of P and d is not straightforward. A heuristic interpretation is however possible by considering the estimates in Table 2. Note that 0i,2'°-$u> '^, 1'2>3, and that 02 is approximately proportional to (0,0,0,1). I'hus, $^,$2 ^nd ^3 can be approximately represented as linear combinations of the vectors ( - 1,1,0,0), (0,0,0,1), and (0,0,1,0), implying that ml-y, i"" and Ap are stationary. This means that the only interesting cointegration relation found is between m 1 and y. However, a linear combination between these three vectors might be more stable (in terms of the roots of the characteristic polynomial) than the individual vectors themselves and this linear combination could in fact be the economically interesting relation. In particular, one would expect that the linear combination, which is most correlated with the stationary part of the model, namely the first eigenvector, is of special interest. Although there is some arbitrariness in the case r > 1, the ordering of tfie eênvectors provided by the estimation procedure is likely to be useful. The estimates reported in Table 2 indicate diat $2 is approximately measuring the inflation rate, whereas $i and ^3 seem to contain information about ml-y. Note also that d,i and d^j have opposite sign. The sign to be expected for 'excess demand for money' sho^d be negative, but di3 dominates ctn, so that the 'excess demand for moliey' enters with a negative sign in the first equation. The value of dj2 can be interpreted as the w e ^ t with which the inflation rate enters equation i. M Table 7 Section VI, the ' It seems reasonable to denote the first coordinate of the cointegraticm vector fii, say, by fin. In ordinary matrix notation we then have fiî - fi^j.


193

estimate of II = o^', i.e. the estimate of the combined effects of all three cointegration vectors, is reported. It is striking how well the proportionality hypothesis between money and income is maintained in all equations of the system. This completes the investigation of the model /fj ^^^ ^* in H^ and we tum now to the models H^ and Hf in / / j . V. ESTIMATION AND TESTING UNDER LINEAR RESTRICTIONS ON P

Mode] 7/3 :/fl = Hflj is a formulation of a linear restriction on the cointegration vectors. The hypothesis spiecifies the same restriction on all the cointegration vectors. The reason for this is the following: If we have two cointegration vectors in which m and y, say, enter then any linear combination of these relations will also be a cointegrating relation. Thus it will in general be possible to find some relation which has, say, equal coefficients with opposite sign to m and y, corresponding to a long-run unit elasticity. This is clearly not interesting, and only if the proportionality restriction is present in all yS vectors, is it meaningful to say that we have found a imit elasticity. 5. /. Likelihood Ratio Tests Under H3 we have the restriction ^ = H 9 where H is (p x j), but that means that the estimation of ri,...,rjt_j, 4>, /i, a and A is given as for fixed jS = Hqj, and q> has to be chosen to minimize I qp'(H'S«H-H'S,oSoo'So,H) flP|/| q>'{n'S,,U) q>\

(5.1)

over the set of al! 5 x r matrices ^. This problem has the same kind of solution as above and we formulate the results in Theorem 5.1 below. A subscript indicates which hypothesis we are currently working with. Throughout, the estimator witiiout subscript will be the estimator under //, orHt THEOREM 5.1: Under the hypothesis wefindthe maximum likelihood estimator of j3 as follows: First solve I AH'S^H - H'SôSo-o'So;tH | = 0,

(5.2)

for i j I >...>i3^ and V3 = (v3i,...,¥3.,) normalized by V3(H'SkjH)V3 = I. Choose * = (*3.,,-,*3.,)and43 = H*, (5.3) and find the estimates of a, A and T fi-om (4.1), (4.2), and (3.5). The maximized likelihood bwomes

194

BULLETIN

which gives the likelihood ratio test of the hypothesis Hj in H2 as

i,Ml - i^l-

(5.5)

The asymptotic distribution of this statistic is shown in Johansen {1989) to be X^ with r{p—s) degrees of freedom. Under the hypothesis H*:fi=Hq> and fi = afi'^, the same results hold. 5.1.1. The Empirical Analysis The Finnish Data We consider the hypothesis that there is proportionality between money and income, so that the coefficients of money and income are equal with opposite sign, i.e. H,:^,,=

-^,,2,

(/= 1,2,3)

In matrix notation the hypothesis can be formulated as: /- 1 1 0 \ 0

0 0 1 0

0 0 0 1

where ^ is a 3 x 3 matrix. Solving (5.2) gives the eigenvalues in Table 4. These are compared to the eigenvalues of the unrestricted model H2. The test statistic is calculated as -21n(2) = 0.02 + 3.51+ 0.29 = 3.82 which is compared to ;u^95(r(/7-s))=;t;^(3(4 —3)) = 7.81. ThiK the hypothesis of equal coefficients with opposite sign for m 1 and y, is clearly accepted. The corresponding restricted ^-estimates hardly change at all compared to the unrestricted estimates of Table 2 and they are therefore not reported here. With the imposed proportionality restriction we now have three cointegration vectors restricted to a three dimensional space defined by the restriction that m 1 and y have equal coefficients with opj)osite sign. Thus the hypothesis if3 is really the hypothesis of a complete specification of sp{p). In this space we can choose to present the results in any basis we want and it seems natural to consider the three variables ml—y, i"' and Ap, Thus the conclusion about the Finnish data is that the last two variables C" and Ap are already stationary, and the first two, y and m 1, are cointegrated. The Danish Data In the Danish data we found r= 1. Based on the imrestricted estimates in the previous section it seems natural to formulate tveo linear hypotheses in this case, both of which are economically meaningful:


195

and In matrix formulation thefirsthypothesis is expressed as 1 - 1 0 0 L 0

0 0

0 0 0 0 1 0 0 0 1 0 0 0 lj

where qj is a 4 x 1 vector. Solving (5.2) gives the eigenvalues in Table 4. These are compared to the eigenvalues of the unrestricted H^ model. The test of //f, in //f consists of comparing Af j and A!f by the test The asymptotic distribution of this quantity is given by the ;f^ distribution with degrees of freedom r{p-s)= 1(4-3)= 1. The test statistic is clearly not significant, and we can thus accept the hypothesis that for the Danish data the coefficients of m2 and y are equal with opposite sign. The second hypothesis that the coefficients for the bond rate and the deposit rate are equal with opposite sign is now tested. This hypothesis implies that the cost of holding money can be measured as the difference between the bond yield and the yield ftom holding money in bank deposits. Since H^, was strongly supported by the data, we will test //J2 within //f,. This will now be formulated in matrix notation as "

1 -1 0 0 0

0 0 1 -1 0

0 0 0 0 1

where y is a 3 x 1 vector. Solving (5.2) we get the eigenvalues reported in Table A. The test for the hypothesis is given by

which should be compared with the x^ quantiles with r\s^ -52)= 1(4-3)= 1 degree of freedom. It is not significant and we conclude the analysis of the cointegration vectors for the Danish demand for money by the restricted estimate iJ' = ( 1.00, -1.00,5.88, -5.88, -6.21), The corresfKMiding estimate of a is given by a* = (-0.177,0.095,0.023,0.032).

196

BUUBTIN

TABLE 4 The Eigenvalues and the Corresponding Test Statistics for Testing Restrictions on fi

',: ',:

Eigenvalues A, 0.309 0.226 0.309 0.199

'f. 't,: '?,:

Eigenvalues Af 0.433 0.178 0.433 0.172 0.423 0.045

0.073 0.070

The Finnish data - 7" ln( 1 - A,) 0.030 38.49 26.64 38.47 23.13

7.89 7.60

3.11

0.113 0.044 0.006

The Danish data —7"ln(l-Af) 0.043 0 30.09 10.36 0.006 30.04 10.01 29.16 2.44

6.34 2.36 0.32

2.35 0.32

0

5.2. The Wald Test

Instead of the likelihood ratio tests which require estimation under the model H2 and 7/3, one can directly apply the results of model H2 given in Table 2 to calculate some Wald tests. The idea is to exprras the restrictions on j8 as K'p=O and then normalize K'/? by its 'standard deviation'. It is shown m Johansen (1989) that if v* denotes the eigenvectors corresponding to A5,..., A^, (see the Danish data in Table 5) then, in case r= 1, the quantity 1/2

r'-i) z is asymptotically Gaussian with mean 0 and variance 1. Hence K* = (K', 0)', such that K*'fi* = K'/3, i.e. the contrast involves oaiy the coefficients of the variables, not the constant term. This statistic is easily calculated from Table 5. If more than one cointegration vector is present, as in the Finnish data, then the Wald statistic is gjven by where v is the eigenvector corresponding to A4, aad 6 = diag(Ai, Aj, ^3) (see the Finnish data in Table 5). The asymptotic distribution of this statistic is x^ with tip-s) degrees of freedom, where K fc p'x.{p-s}. In this case r=p-l = 3, and since r^s^p = 4 and, since s»p is no restriction, we can only test a hypothesis with s = r=p —1-3, coiresponding to a completely specified fi. The above test statistics require the nonnaUza^n of fi and v as in (4.5). An alternative expression for this statistic which can be ^^lied for any normalization is


197

5.2.1. The Empirical Analysis Since the calctilations are numerically simpler for the normalization VStiV = I, it will be tised to illustrate the Wald tests. In Table 5 the eigenvalues and eigenvectors for this normalization Eire reported. 7?ie Danish Data We start by the hypothesis expressed as K'p = {1,1,0,0) P = 0. The Wald statistic is then calculated as foUows: First, TÎ^K'p = 53^l\ -21.97 + 22.70) = 5.31, and 5

I (K*'v*)2 = ( 14.66-20.05)2+ (7.95-25.64)2 Then the test statistic becomes w = 5.31/(1/0.4332-l)x 360.13)'/-= 0.24. The second hypothesis is tested in a similar way. Note however, that //f 2 is now tested within Hf and not within /ff,. The test statistics becomes 1.32. Both these statistics are asymptotically normalized Gaussian and the values found are hence not significant. The Finnish Data For the Finnish data we only test the hypothesis: ^3:^.1=-/3,2,

(i = 1,2,3).

This can be formulated as K'y? = (1,1,0,0) P = 0. First wefindfromTable 5 that K'w'K=(1.38 + 2.22)^ = 12.96 and

^ ( - 1 1 . 1 3 + 10.24)-^^_^3. 0.0731""' - 1 The test statistic becomes w^= 104x0.83/12.96 = 5.66, which is not sipifiramt in the ;f ^ distrilwtion with 3 degrees of freedom. Notice that the Wald test in aO cases givra a value of the test statistic which is lar^r tijan the value for the likelihood ratio test statistic. This just

198

BUU£TIN

(S

q d

O

.d

* ^ ^H

1:

ts

^

2 - : d

Q g H "^ U| d

•»•

r~

o

-^

ts

-cj-'

fN)

(^

•^^

fs : H^K

ts I

INFERENCE ON COINTEGRATiON

199

emphasizes the feet that we are relying on asymptotic results and a careful study of the small sample properties is needed. VI. ESTIMATION AND TESTING UNDER RESTRICTIONS ON a

Let US now turn to the hypothesis H4 where a is restricted by o = A V in the model H2. Here A is a (pxm) matrix. It is convenient to introduce B{px{p-m)) = Ai, such that B'A = O. Then the hypothesis H^, can be expressed as B'a = 0. The concentrated likelihood function (3.8) can be expressed in the variables given by A'(Ro, - a/J'R,,) = A'Ro, - A'A V^'R^,

(6.1) (6.2)

In the following, we factor out that part of the likelihood function which depends on B'Rg,, since it does not contain the parameters V and fi. To save notation, we define: A^=A'AA, A^j = A'AB, S^^^ = S„^ - S^^S^ft'Sî = A'So, - A'SooB(B'SooB)- 'B'So*, etc. The factor corresponding to the marginal distribution of B'R^, is given by

- Z (B'Ro,)'A^,HB'Ro,)/2L

(6.3)

and gives the estimate A,, = S,, = B'SooB,

(6.4)

and the maximized likelihood function from the marginal distribution (6.5) The other factor corresponds to the conditional distribution of A'R^, and Rj, conditional on B'RQ, and is given by

X A-',(A'Ro, - A'A#'R,, - A«,A,VB'Ro,)/2 j .

(6.6)

It is a well-known result from the Aeoiy of the midtivariate normal distribution that the parameters Aji,^, AÂ^^^ Mid A^^^ are variation independent and hence that the esdtnate of A ^jÂ ^' is found by regression for fixed ^ and

200

BULUETIN

/3 giving KJ^-M'(

V, ;»)=(S,, - A'AVJS'S,,) S^,',

(6.7)

and new residuals defined by

In terms offtândft^the concentrated likelihood function has the form (3.8) which means that the estimatiOTi of /J follows as before.* THEOREM 6.1: Under the hypothesis the maximum likelihood estimator of fi is found as follows: First solve the equation |AS,,.,-St,.,S;^',S«,.,| = O, (6.8) giving i^.i >...> A4.«>i4^.,, = ... = i4.p = 0 and^4 = (V4,,...,V4,,) normalized suchthatSi'^S^ti (,^4 = 1. Now take k = {Kl Kr\ (6-9) which gives the estimates ^=(A'A)-'S,,.^4

(6-10)

and ,«.*

oB(B'SooB)-iB'So,)44, = S^,,,-A'd4d;A,

,,.^

(6.11) (6.12)

and the maximized likelihood function "' (6.13) The estimate of A can be foimd from (6.4), (6.7) and (6.12), and T is estimated from (3.5). The likelihood ratio test statisic of H^ in / / , is - 2 hi(Q; H41 Hj) = r I ln!( 1 - k,M 1 - A,)}.

(6.14)

1-1

The asymptotic distribution of this test statistic is pven by a x^ distribution wiA r{p-m) degrees of freedom, see Johansiai (1989). The same result holds for testing HJ: a = A ^ in /f f. Ît is convenient to calctdate the rdevant product momem matrices as


201

The following very simple CoroUmy is usefiil for explaining the role of single ûation analysis: COROLLARY 6.2: If m = r = 1 then the maximum likelihood estimate of fi is found as the coefficients of X,_^ in the regression of A'AX, on X,_j, B'AX,, and AX, _ 1,..., AX, _ ;t +1, D, and the constant. PROOF: It suffices to notice that when m = r = 1, then only one cointegration vector has to estimated. It is seen from (6.8) that since the matrix Sna./rSaa'aSat ft is singular and in feet of rank 1, then only one eigenvalue is non-zero, and the corresponding eigenvector is proportional to SZk,t^ka.b-> which is exactly the regression coefficient of R^, obtained by regressing A'R^,, on B'Rfl, and R^. This can of course be seen directly from (6.6) since A'A^ is 1x1 and can be absorbed into fi, which shows that fi is given by the regression as described. If, in particular, a is proportional to (1,0,...,0), then ordinary least squares analysis of the first equation will give the maximum likelihood estimation for the cointegration vector. An empirical illustration of this will be presented below. Finally we just state briefly how one solves the estimation and testing of the model H^.fi = Yl
-r.In case the processF isgivenbyU-U, see (4.11) the stochastic matrix jFF'dt and JF dU' are approximated by

r-l

and

respectively, where X_, = I ~' Z X, _,. From these expressions we calculate

From this matrix flie trace and the maximum eigenvalue are calculated. On the basis of 6,000 simulations the quantiles are found as the appropriate order statistics. If instead F is given by (4.14) and (4.15), we replace in the above calculation the last component of X,_ j - X.., by r - 1 / 2 , and if F is given by (4.16) and (4.17) then X_| is droppisd and X,_i is extended by an extra component 1.

208

BULLETIN TABLE A1 Distribution ofthe Maximal Eigenvalue and Trace ofthe Stochastic Matrix |(dU)F'[JFF'dw]-'JF(dtr') where U is an wi-dimensional Brownian motion and F is U - C, except that the last component is replaced hy t- 1/2, see Theorem 4.1 dim 50%

80%

90%

1. 2. 3. 4. 5.

0.447 6.852 12.381 17.719 23.211

1.699 10.125 16.324 22.113 27.899

2.816 12.099 18.697 24.712 30.774

3.962 14.036 20.778 27.169 33.178

1. 2. 3. 4. 5.

0.447 7.638 18.759 33.672 52.588

1.699 11.164 23.868 40.250 60.215

2.816 13.338 26.791 43.964 65.063

3.962 15.197 29.509 47.181 68.905

95%

97.5%

99%

mean

var

5.332 15.810 23.002 29.335 35.546

6.936 17.936 25.521 31.943 38.341

1.030 7.455 12.951 18.275 23.658

2.192 12.132 18.549 23.837 28.330

5.332 17.299 32.313 50.424 72.140

6.936 19.310 35.397 53.792 76.955

1.030 8.250 19.342 34.184 52.998

2.192 14.065 32.103 55.249 82.106

Maximal eigenvalue

Trace

Simulations are performed replacing the Brownian motion by a Gaussian random walk with 400 steps and the process is stimulated 6.000 times. TABLE A2

Distribution ofthe Maximal Eigenvalue and Trace ofthe Stochastic Matrix J(dU)F'[|FF'd«]-'|F(dU') where U is an wi-dimensional Brownian motion and F = U — U dim 50%

m%

90%

1. 2. 3. 4. 5.

2.415 7.474 12.707 17.875 23.132

4.905 10.666 16.521 22.341 27.953

6.691 12.783 18.959 24.917 30.818

8.083 14.595 21.279 27.341 33.262

1. 2. 3. 4. 5.

2.415 9.335 20.188 34.873 53.373

4.905 13.038 25.445 41.623 61.566

6.691 15.583 28.436 45.248 65.956

8.083 17.844 31.256 48.419 69.977

95%

99%

mean

var

9.658 16.403 23.362 29.599 35.700

11.576 18.782 26.154 32.616 38.858

3.030 8.030 13.278 18.451 23.680

7.024 12.568 18.518 24.163 29.000

9.658 19.611 34.062 51.801 73.031

11.576 21.962 37.291 55.551 77.911

3.030 9.879 20.809 35.475 53.949

7.024 18.017 34.159 56.880 84.092

97.5%

Maximal eigenvalue

Trace

Simulauons are perfonned t^ reptacirtg the Brownian roodeti by a Gaussian random walk with 400 steps and the process is simulated 6,000 times.


209

TABLE A3

Distribution ofthe Maximal Eigenvalue and Trace ofthe Stochastic Matrix j{dV)F'[j¥F'du]-'j¥{dV') where U is an m-dimensional Brownian motion and F is an (m +1 )-diniensiona! process equal to U extended by 1, see Theorem 4.1 dim 50%

80%

90%

95%

99%

mean

var

10.709 17.622 23.836 30.262 36.625

12.740 19.834 26.409 33.121 39.672

4.068 8.917 14.050 19.172 24.433

6.738 13.021 18.698 23.607 28.954

10.709 22.202 37.603 56.449 78.857

12.741 24.988 40.198 60.054 82.969

4.068 12.017 23.868 39.431 58.954

6.738 19.192 37.529 59.854 89.072

97.5%

Maximal eigenvalue 1. 2. 3. 4. 5.

3.474 8.337 13.494 18.592 23.817

5.877 11.628 17.474 22.938 28.643

7.563 13.781 19.796 25.611 31.592

9.094 15.752 21.894 28.167 34.397 Trace

1. 2. 3. 4. 5.

3.474 11.381 23.243 38.844 58.361

5.877 15.359 28.768 45.635 66.624

7.563 17.957 32.093 49.925 71.472

9.094 20.168 35.068 53.347 75.328

Simulations are performed by replacing the Brownian motion by a Gaussian random walk with 400 steps and the process is simulated 6,000 times.

REFERENCES

Ahn, S. K. and Reinsel, G. C. (1987). 'Estimation for Partially Non-stationary Multivariate Autoregressive Models', University of Wisconsin. Anderson, T. W. (1984). An Introduction to Multivariate Statistical Analysis. New YortWUey. Box, G. E. P. and Tiao, G. C. (1981). 'A Canonical Analysis of Multiple Time Series with Applications', Biometrika, Vol. 64, pp. 355-65. Davidson, J. (1986). 'Cointegration in Linear Dynamic Systems', Discussion paper, LSE. Dolado, J. J. and Jenkinson, T. (1988). 'Cointegration: A Survey of Recent Developments', Mimeo, Institute of Economics and Statistics, Oxford University. Engle, R F. and Granger, C. W. J. (1987). Co-integration and Error Correction: Representation, Estimation and Testing', Econometrica, Vol. 55, pp. 251-76. Fuller, W. A. (1976). Introduction to Statistical Time Series, New York, Wiley. Granger, C. W. J. (1981). 'Some Properties of Time Series Data and their Use in Econometric Model Specification', ytM